We explore whether supermassive non-baryonic stars (in particular boson, mini-boson and non-topological soliton stars) might be at the center of some galaxies, with special attention to the Milky Way. We analyze, from a dynamical point of view, what current observational data show, concluding that they are compatible with a single supermassive object without requiring it to be a black hole. Particularly, we show that scalar stars fit very well into these dynamical requirements. The parameters of different models of scalar stars necessary to reproduce the inferred central mass are derived, and the possible existence of boson particles with the adequate range of masses is commented. Accretion to boson stars is also analyzed, and a comparison with another non-baryonic candidate, a massive neutrino ball, which is also claimed as an alternative to the central black hole, is given. Both models are capable to explain the nature of the object in Sgr A$^*$ without invoking the presence of a singularity. One difficult issue is why the accreted materials will not finally produce, in a sufficiently long time, a black hole. We provide an answer based on stellar disruption in the case of boson stars, and comment several suggestions for its possible solution in neutrino ball scenarios. Finally, we discuss the prospects for the observational detection of these supermassive scalar objects, using the new generation of X-ray and radio interferometry satellites.

22 double column pages, 10 separate eps figures